Problem: $\dfrac{ g - 3h }{ -7 } = \dfrac{ -9g + 5i }{ -10 }$ Solve for $g$.
Solution: Multiply both sides by the left denominator. $\dfrac{ g - 3h }{ -{7} } = \dfrac{ -9g + 5i }{ -10 }$ $-{7} \cdot \dfrac{ g - 3h }{ -{7} } = -{7} \cdot \dfrac{ -9g + 5i }{ -10 }$ $g - 3h = -{7} \cdot \dfrac { -9g + 5i }{ -10 }$ Multiply both sides by the right denominator. $g - 3h = -7 \cdot \dfrac{ -9g + 5i }{ -{10} }$ $-{10} \cdot \left( g - 3h \right) = -{10} \cdot -7 \cdot \dfrac{ -9g + 5i }{ -{10} }$ $-{10} \cdot \left( g - 3h \right) = -7 \cdot \left( -9g + 5i \right)$ Distribute both sides $-{10} \cdot \left( g - 3h \right) = -{7} \cdot \left( -9g + 5i \right)$ $-{10}g + {30}h = {63}g - {35}i$ Combine $g$ terms on the left. $-{10g} + 30h = {63g} - 35i$ $-{73g} + 30h = -35i$ Move the $h$ term to the right. $-73g + {30h} = -35i$ $-73g = -35i - {30h}$ Isolate $g$ by dividing both sides by its coefficient. $-{73}g = -35i - 30h$ $g = \dfrac{ -35i - 30h }{ -{73} }$ Swap signs so the denominator isn't negative. $g = \dfrac{ {35}i + {30}h }{ {73} }$